Equivalential Algebras with Conjunction on Dense Elements





intuitionistic logic, Fregean varieties, equivalential algebras, dense elements


We study the variety generated by the three-element equivalential algebra with conjunction on the dense elements. We prove the representation theorem which let us construct the free algebras in this variety.


J. Czelakowski, D. Pigozzi, Fregean logics, Annals of Pure and Applied Logic, vol. 127(1–3) (2004), pp. 17–76, DOI: https://doi.org/10.1016/j.apal.2003.11.008
Google Scholar DOI: https://doi.org/10.1016/j.apal.2003.11.008

R. Freese, R. McKenzie, Commutator theory for congruence modular varieties, vol. 125 of London Mathematical Society Lecture Notes, Cambridge University Press, Cambridge (1987).
Google Scholar

J. Hagemann, On regular and weakly regular congruences, Tech. rep., TH Darmstadt (1973), preprint no. 75.
Google Scholar

D. Hobby, R. McKenzie, The structure of finite algebras, vol. 76 of Contemporary Mathematics, American Mathematical Society (1988), DOI: https://doi.org/10.1090/conm/076
Google Scholar DOI: https://doi.org/10.1090/conm/076

P. M. Idziak, K. Słomczyńska, Polynomially rich algebras, Journal of Pure and Applied Algebra, vol. 156(1) (2001), pp. 33–68, DOI: https://doi.org/10.1016/S0022-4049(99)00119-X
Google Scholar DOI: https://doi.org/10.1016/S0022-4049(99)00119-X

P. M. Idziak, K. Słomczyńska, A.Wroński, Fregean varieties, International Journal of Algebra and Computation, vol. 19(5) (2009), pp. 595–645, DOI: https://doi.org/10.1142/S0218196709005251
Google Scholar DOI: https://doi.org/10.1142/S0218196709005251

P. M. Idziak, K. Słomczyńska, A. Wroński, The commutator in equivalential algebras and Fregean varieties, Algebra Universalis, vol. 65(4) (2011), pp. 331–340, DOI: https://doi.org/10.1007/s00012-011-0133-4
Google Scholar DOI: https://doi.org/10.1007/s00012-011-0133-4

J. K. Kabziński, A. Wroński, On equivalential algebras, [in:] G. Epstein, J. M. Dunn, S. C. Shapiro, N. Cocchiarella (eds.), Proceedings of the 1975 International Symposium on Multipe-Valued Logic, Indiana University, Bloomington, Indiana, IEEE Computer Society, Long Beach (1975), pp. 419–428, URL: https://apps.dtic.mil/sti/pdfs/ADA045757.pdf
Google Scholar

P. Köhler, Brouwerian semilattices, Transactions of the American Mathematical Society, vol. 268(1) (1981), pp. 103–126, DOI: https://doi.org/10.1090/S0002-9947-1981-0628448-3
Google Scholar DOI: https://doi.org/10.2307/1998339

R. McKenzie, G. McNulty, W. Taylor, Algebras, Lattices, Varieties: Volume I, AMS Chelsea Publishing, Providence, Rhode Island (1987).
Google Scholar

S. Przybyło, Equivalential algebras with conjunction on the regular elements, Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, vol. 20 (2021), pp. 63–75, DOI: https://doi.org/10.2478/aupcsm-2021-0005
Google Scholar DOI: https://doi.org/10.2478/aupcsm-2021-0005

H. Rasiowa, R. Sikorski, The Mathematics of Metamathematics, vol. 125, PWN, Warszawa (1963).
Google Scholar

K. Słomczyńska, Equivalential algebras. Part I: representation, Algebra Universalis, vol. 35(4) (1996), pp. 524–547, DOI: https://doi.org/10.1007/BF01243593
Google Scholar DOI: https://doi.org/10.1007/BF01243593

K. Słomczyńska, Free spectra of linear equivalential algebras, The Journal of Symbolic Logic, vol. 70(4) (2005), pp. 1341–1358, DOI: https://doi.org/10.2178/jsl/1129642128
Google Scholar DOI: https://doi.org/10.2178/jsl/1129642128

K. Słomczyńska, Free equivalential algebras, Annals of Pure and Applied Logic, vol. 155(2) (2008), pp. 86–96, DOI: https://doi.org/10.1016/j.apal.2008.03.003
Google Scholar DOI: https://doi.org/10.1016/j.apal.2008.03.003

K. Słomczyńska, Unification and projectivity in Fregean varieties, Logic Journal of the IGPL, vol. 20(1) (2011), pp. 73–93, DOI: https://doi.org/10.1093/jigpal/jzr026
Google Scholar DOI: https://doi.org/10.1093/jigpal/jzr026

K. Słomczyńska, The structure of completely meet irreducible congruences in strongly Fregean algebras, Algebra universalis, vol. 83 (2022), DOI: https://doi.org/10.1007/s00012-022-00787-0 article number: 31.
Google Scholar DOI: https://doi.org/10.1007/s00012-022-00787-0

M. H. Stone, The theory of representation for Boolean algebras, Transactions of the American Mathematical Society, vol. 40(1) (1936), pp. 37–111, DOI: https://doi.org/10.2307/1989664
Google Scholar DOI: https://doi.org/10.2307/1989664

A. Wroński, On the free equivalential algebra with three generators, Bulletin of the Section of Logic, vol. 22 (1993), pp. 37–39.
Google Scholar




How to Cite

Przybyło, S., & Słomczyńska, K. (2022). Equivalential Algebras with Conjunction on Dense Elements. Bulletin of the Section of Logic, 51(4), 535–554. https://doi.org/10.18778/0138-0680.2022.22



Research Article